Polar Coordinates and Complex Numbers
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We're using the distance formula in polar coordinates. This problem asks us to plot these two points, and find the distance between them. First let's plot 6, 4 pi over 3.
Now I have to identify the angle 4 pi over 3. Well, that's the same as 8 pi over 6. Pi over 6 is 30 degrees, that's 1, 2, 3, 4, 5, 6, 7, 8 and this is 8 pi over 6, or 4 pi over over 3. I need to go six units along in this direction. So 1, 2, 3, 4, 5, 6, that's my first point.
My second point; 8, negative pi over 6. The angle negative pi over 6, remember negative angles are measured clockwise. This is the direction negative pi over 6 and 8 in this direction will take me 1, 2, 3, 4, 5, 6, 7, 8, to this point. I want to find the distance between these two points. I'll draw the distance between them as a segment. I want to find the length of this segment.
That takes me to the distance formula in polar coordinates. Distance squared equals r1² plus r2² minus 2 times r1 times r2 times the cosine of theta 2 minus theta 1.
What's the distance between these two points? Well, d² equals, I'll call this r1, so r1² is 36, plus r2² 64, minus 2 times r1 times r2, 6 times 8, times the cosine of the angle between them. The angle between them is 4 pi over 3 minus negative pi over 6.
Now one interesting thing about when you calculate the angle between them, the order that you subtract doesn't matter. The cosine is going to be the same for this angle minus this angle, and this angle minus this angle, because cosine is an even function. So you don't have to worry about the order that you subtract.
So this is 4 pi over 3 minus as I said before is 8 pi over 6. Now 4 pi over 3 is as I said before 8 pi over 6. I'm adding pi over 6, so this is 9 pi over 6 which is 3 pi over 2. So I have 100 minus 48 times 2, 96, times the cosine of 3 pi over 2. Now the cosine of 3 pi over 2 is 0, so this is just 100. This is 0. So we get d² is 100, and that means that d is 10.
Now what does it mean for the cosine of the angle between to be 0? This angle is actually a right angle. So if we had known that, we had observed that, we could have used the Pythagorean theorem, but it doesn't matter. You don't have to be aware of the actual value of the angle. Just use the formula, and it will take care of the rest. The distance between the two points is 10.